Numerus defectivus

Numerus defectivus est numerus ${\displaystyle n}$, cuius summa divisorum minor numero ${\displaystyle n}$ ipso - e summa excepto - est. Vel symbolis mathematicis expressum

Systemata Numerica Mathematicae

Numeri Elementarii
Naturales ${\displaystyle \mathbb {N} }$ {0,1,2,3,...} sive {1,2,3,...} Primi ${\displaystyle \mathbb {P} }$ {2,3,5,7,11,...} Abundantes Amicabiles Compositi Defectivi Perfecti Sociabiles Integri ${\displaystyle \mathbb {Z} }$ {...,-2,-1,0,+1,+2,...} Pares {...,-2,0,+2,...} Impares {...,-3,-1,+1,+3,...} Rationales ${\displaystyle \mathbb {Q} }$ Reales ${\displaystyle \mathbb {R} }$ Irrationales Numerus pi ${\displaystyle \pi =3.14159265358979\ldots }$ Numerus Euleri ${\displaystyle e=2.718281828459045\ldots }$ Complexi ℂ Algebraici Transcendentes Numerus imaginarius ${\displaystyle i={\sqrt {-1}}}$ Quaterni ${\displaystyle \mathbb {H} }$ Octoni ${\displaystyle \mathbb {O} }$ Infinitas ${\displaystyle \infty }$
Variae radices
Radix binaria(2) Radix octalis(8) Radix decimalis(10) Radix sedecimalis(16)

${\displaystyle \sum _{d|n,d\not =n}d<n}$.

Exempla:

• 10 est numerus defectivus, nam 1+2+5 = 8 < 10.
• Omnes numeri primi sunt defectivi, cum 1 sit eorum solus divisor, qui summetur (bene observes: numerus ${\displaystyle n}$ ipse semper e summa exceptus est).

Si summa divisorum numeri ${\displaystyle n}$ uno est minor ${\displaystyle n}$ ipso, ${\displaystyle n}$ appellatur numerus leviter^? defectivus. Cunctae potentiae numeri 2 sunt leviter defectivae:

Potentia	Summa divisorum
${\displaystyle 2^{2}=4}$	${\displaystyle 1+2=3}$
${\displaystyle 2^{3}=8}$	${\displaystyle 1+2+4=7}$
${\displaystyle 2^{4}=16}$	${\displaystyle 1+2+4+8=15}$
...	...
${\displaystyle 2^{n}}$	${\displaystyle 2^{n}-1}$ [1]

Nexus interni

• Numerus abundans
• Numerus perfectus